Correlation and variance.

Question

If the correlation between two variables is 0.50, then how much variance does X account for Y? It seems very simple, but I can't actually get the point of the question, does it require us to calculate $\frac{\sigma_x^2}{\sigma_y^2}$?

Answer 1

$$ \\ \sigma^2_{Y} = \beta^2 \sigma^2_{X} + \sigma^2_{\epsilon} \\ $$

$$\beta = \frac{\sigma_{X,Y}}{\sigma^2_{X}} = \rho_{X,Y}\frac{\sigma_{Y}}{\sigma_{X}}$$

$$\beta^2 =.5^2\frac{(\sigma_{Y})^2}{(\sigma_{X})^2}$$

$$ \sigma^2_{Y} = .25 \times \sigma^2_{Y} + \sigma^2_{\epsilon} $$

Thus a correlation of $0.5$ means X accounts for $25$% of the variance of Y